A complete set of invariants for finite groups and other results

on some invariants of finite groups

in this note we are going to survey several invariants of finite groups related either to their orders or to generating sets or to lattices of subgroups. some relations among these invariants will be exhibited. special attention will be paid to monotonicity of them.

Constructing Invariants for Finite Groups

A Complete Set of Unitary Invariants for 3x3 Complex Matrices

is a complete set of unitary invariants for any wX» complex matrix A. The author was able to improve this result by demonstrating in [3] that for re fixed but arbitrary, there is always a subset of the above collection containing fewer than 24" traces which serves as a complete set of unitary invariants for «X» matrices. It is of interest to discover the sharpness of the above upper bound on th...

Some numerical results on two classes of finite groups

In this paper, we consider the finitely presented groups $G_{m}$ and $K(s,l)$ as follows;$$G_{m}=langle a,b| a^m=b^m=1,~[a,b]^a=[a,b],~[a,b]^b=[a,b]rangle $$$$K(s,l)=langle a,b|ab^s=b^la,~ba^s=a^lbrangle;$$and find the $n^{th}$-commutativity degree for each of them. Also we study the concept of $n$-abelianity on these groups, where $m,n,s$ and $l$ are positive integers, $m,ngeq 2$ and $g.c.d(s,...

Finite Simple Groups and Dickson Invariants

In a recent paper [1], Adem, Maginnis and Milgram calculated the mod 2 cohomology of the Mathieu simple group M12. An interesting feature of the answer they obtain is that it is Cohen–Macaulay. There is a subalgebra isomorphic to the rank three Dickson invariants, namely a polynomial ring in generators of degrees 4, 6 and 7, and over this subalgebra the whole cohomology ring is a finitely gener...